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Theorem bj-1uplex 36991
Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-1uplex (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)

Proof of Theorem bj-1uplex
StepHypRef Expression
1 bj-pr11val 36988 . . 3 pr1𝐴⦆ = 𝐴
2 bj-pr1ex 36989 . . 3 (⦅𝐴⦆ ∈ V → pr1𝐴⦆ ∈ V)
31, 2eqeltrrid 2844 . 2 (⦅𝐴⦆ ∈ V → 𝐴 ∈ V)
4 df-bj-1upl 36981 . . 3 𝐴⦆ = ({∅} × tag 𝐴)
5 p0ex 5390 . . . 4 {∅} ∈ V
6 bj-xtagex 36972 . . . 4 ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V))
75, 6ax-mp 5 . . 3 (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)
84, 7eqeltrid 2843 . 2 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
93, 8impbii 209 1 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  Vcvv 3478  c0 4339  {csn 4631   × cxp 5687  tag bj-ctag 36957  bj-c1upl 36980  pr1 bj-cpr1 36983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-bj-sngl 36949  df-bj-tag 36958  df-bj-proj 36974  df-bj-1upl 36981  df-bj-pr1 36984
This theorem is referenced by:  bj-2uplex  37005
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